SEO: target "product launch videos" alongside "startup launch videos", paginate categories, real lastmod

[dingyi]

Why

The site copy only ever said "startup launch videos"; "product launch video" appeared solely in fallback strings. Only the first 60 entries of the homepage and of each category were linked from static HTML, leaving ~2,100 entry pages reachable only via the sitemap. dateModified was a hardcoded constant on every page and the sitemap had no <lastmod>.

What changes

Keywords (both phrases kept, no layout changes)

• Homepage <title> → Startup & Product Launch Videos from X — What Ships; meta description, hero copy, crawler intro and JSON-LD names now carry both phrases.
• Category pages: title {Label} startup & product launch videos — What Ships, H1 unchanged, copy mentions product launch videos.
• Entry pages: title suffix — Product launch video | What Ships (motion entries: Launch film); definition sentence says "product launch video"; VideoObject.genre added.

Internal linking

• PAGE_SIZE 60 → 100: homepage HTML now links 100 entries.
• Category listings are static paginated routes: /videos/category/{id}/ + /videos/category/{id}/{n}/, with rel=prev/next, every page number linked from every page (max 9 pages), and matching markdown siblings. Replaces the "showing 60 of N, see home grid" paragraph, styled like the existing category chips.
• Entry pages link their category (fact row + prose) and the BreadcrumbList gains a category level.

Freshness / structured data

• dateModified derives from the catalog: per entry, post date or newer view snapshot; per listing, newest entry change. Sitemap emits <lastmod> for every catalog-backed URL (hand-written pages omit it rather than claim a date).
• Homepage JSON-LD now actually receives the newest-entries ItemList it was written to emit.
• Drop VideoObject.embedUrl (pointed at the HTML page, not a player).

Data fix

• Two published entries used category devtools (not in the taxonomy) and appeared on no category page; moved to developer-tools, with a test guarding unknown ids.

Verification

• vitest run: 106 passed (new tests for date helpers, sitemap entries, paginated category markdown, category ids).
• astro check: 0 errors. astro build: 2,198 pages; sitemap has 2,199 URLs / 2,189 lastmods; /videos/category/ai/2/ … /9/ exist with 100 links each.

Category pagination, desktop
Category pagination, mobile

Not in this PR

• The editorial landing page for "product launch video examples" (proposed as a separate PR).
• Making the hidden homepage intro visible (left as-is to preserve layout).
• Cloudflare bot-challenge check: verify in Search Console that Googlebot receives 200 for /.

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